Let's compare 0.1 to 0.070. So this 1 right over here,
it is in the tenths place. So it literally
represents 1 times 1/10, which is obviously the
same thing as 1/10. Now, when we look at this
number right over here, it has nothing in
the tenths place. It has 7 in the
hundredths place. So this is the hundredths
place right over here. And then it also has nothing
in the thousandths place. So this number can be rewritten
as 7 times 1/100, or 7/100. And now we could compare
these two numbers. And there's two ways you
could think about it. You could try to turn
1/10 into hundredths. And the best way to do that, if
you want the denominator to be increased by a factor
of 10, you need to do the same thing
to the numerator. So all I did is I multiplied
the numerator and denominator by 10. Ten 100's is the exact
same thing as 1/10. And here it becomes
very clear, 10/100 is definitely larger than 700. Another way you could
think about this is, look, if you were to
increment by hundredths here, you would start at
7/100, 8/100, 9/100, and then you would
get to 10/100. So then you would
get to that number. So this number, multiple ways
you could think about it, is definitely larger. So let me write this down. This is definitely
larger, greater than. This is greater than that. The greater than symbol
opens to the larger value. So here we have 0.093
and here we have 0.01. So let's just think
about this a little bit. So this 9-- get
a new color here. This 9 is not in the
tenths, the hundredths. It's in the thousandths place. It's in the thousands place. And this 3 is in the-- I'm
running out of colors again. This 3 is in the ten
thousandths place. So the 3 is in the
ten thousandths place. So you could literally view
this as 9/1,000 plus 3/10,000. And if you just wanted to write
it in terms of ten thousandths, you can multiply the
9 and 1,000 by 0. And so it becomes 90/10,000. And if you want to add them
together, you could, of course, write this as 93/10,000. Ten thousandths. I always have trouble with
that "-ths" at the end. Now, let's think about this
number right over here, 0.01. Well, this 1 right over here
is in the hundredths place. It's in the hundredths place. So it literally
represents 1/100. So how can we compare
1/100 to 93/10,000? So the best way
to think about it is, well, what's 1/100 in
terms of ten thousandths? Well, let's just multiply
both the numerator and the denominator
here by 10 twice. Or you could say, let's
multiply them both by 100. If you multiply by 10 once,
you get to ten thousandths. It's the same thing as 1/100. Multiply by 10 again, you get
100/10,000 is the same thing as 1/100. And we know that, 100
times 100 is 10,000. So here it becomes very
clear, 100/10,000, or 1/100, is definitely a
larger than 93/10,000. So this quantity
right over here is less than this quantity there. Less than symbol, the small end
points to the smaller number, larger end to the larger number. In fact, that's true with the
less than and greater than. So let's see, this one right
over here, 0.6 versus 0.06. So here, I have a 6
in the tenths place. So it literally represents 6/10. And in the second, I have a
6 in the hundredths place. Well, 6/100 is definitely
smaller than 6/10. A hundredth is a
tenth of a tenth. So this one is pretty
straightforward. This is going to be
the larger value. 0.6 is greater than 0.06. Now, let's think
about 0.3 versus 0.06. So this 3 literally
represents 3/10 while this 6 right over
here represents 6/100. And if you wanted to
compare them directly, you could multiply 3/10 times--
well, both the numerator and the denominator by 10 so
you're not changing its value. 10/10 is essentially
1, or it is 1. So this becomes 30/300. 3/10 is the same
thing as 30/100. And 30/100 is a lot
larger than 6/100. So this is greater than.